Question:
Using the normalization integration for a Gaussian random variables, find an analytic expression (closed-form solution) for the following integral $I=\int^{\infty}_{-\infty}e^{-(ax^2+bx+c)}dx$
Solution:
$e^{-(ax^2+bx+c)}=e^{-a(x^2+\frac{b}{a}x+(\frac{b}{2a})^2+c-\frac{b^2}{4a^2})}=e^{-a(x+\frac{b}{2a})^2} \times {e^{\frac{b^2}{4a}-c}}$,so
$\int^{\infty}_{-\infty}e^{-(ax^2+bx+c)}dx=e^{\frac{b^2}{4a}-c}\times \sqrt{\frac{2\pi}{2a}} \int^{\infty}_{-\infty}\frac{1}{\sqrt{\frac{2\pi}{2a}}}e^{-\frac{(x+\frac{b}{2a})^2}{\frac{1}{a}}}dx=e^{(\frac{b^2}{4a}-c)} \times \sqrt{\frac{\pi}{a}}$
For these formula ,I think it is not logical for me,i mean,i think lots of people have to calculate by rote to solve or calculate this question,is there more logical way to solve this?i mean,by definition,and solve it,not calculate it by rote.